Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-11T00:08:52.160Z Has data issue: false hasContentIssue false
  • English
  • Français

A notion of homotopy for the effective topos

Published online by Cambridge University Press:  17 November 2014

JAAP VAN OOSTEN
JAAP VAN OOSTEN*
Affiliation:
Department of Mathematics, Utrecht University, P.O. Box 80.010, 3508 TA Utrecht, the Netherlands E-mail: J.vanOosten@uu.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define a notion of homotopy in the effective topos.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Special Issue 5: From type theory and homotopy theory to Univalent Foundations of Mathematics , June 2015 , pp. 1132 - 1146
Copyright
Copyright © Cambridge University Press 2014 

References

Awodey, S. and Warren, M. (2009) Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1) 45–55.Google Scholar
Baues, H. J. (1989) Algebraic homotopy, Cambridge studies in advanced mathematics volume 15, Cambridge University Press.CrossRefGoogle Scholar
Dwyer, W. G. and Spalinski, J. (1995) Homotopy theories and model categories. In: James, I. M. (ed.) Handbook of Algebraic Topology, Elsevier.Google Scholar
Gambino, N. and Garner, R. (2008) The identity weak factorization system. Theoretical Computer Science 409 (1) 94–109.Google Scholar
Hirschhorn, P. S. (2003) Model Categories and Their Localizations, Mathematical Surveys and Monographs volume 99, AMS.Google Scholar
Hovey, M. (1999) Model Categories, Mathematical Surveys and Monographs volume 63, AMS.Google Scholar
Hyland, J. M. E. (1982) The effective topos. In: Troelstra, A. S. and Van Dalen, D. (eds.) The L.E.J. Brouwer Centenary Symposium, North Holland Publishing Company 165–216.Google Scholar
Hyland, J. M. E., Robinson, E. P. and Rosolini, G. (1990) The discrete objects in the effective topos. Proceedings of the London Mathematical Society 60 1–60.Google Scholar
Kapulkin, C., Lumsdaine, P. L. and Voevodsky, V. (2012) The simplicial model of univalent foundations. Available at: arXiv:1211.2851v1.Google Scholar
Quillen, D. G. (1967) Homotopical Algebra, Lecture Notes in Mathematics volume 43, Springer.Google Scholar
Strøm, A. (1972) The homotopy category is a homotopy category. Archiv der Mathematik XXIII 435–441.CrossRefGoogle Scholar
The Univalent Foundations Program. (2013) Homotopy Type Theory, Institute for Advanced Study, Princeton. Collectively written account of HoTT by participants of the Special Year on Univalent Foundations of Mathematics, organized by S. Awodey, Th. Coquand and V. Voevodsky.Google Scholar
van den Berg, B. and Garner, R. (2011) Types are weak ω-groupoids. Proceedings of the London Mathematical Society 102 (3) 370–394.Google Scholar
van den Berg, B. and Garner, R. (2012) Topological and simplicial models of identity types. ACM Transactions on Computational Logic (TOCL) 13 (1) 1–44.Google Scholar
van Oosten, J. (2008) Realizability: An Introduction to its Categorical Side, Studies in Logic volume 152, North-Holland.Google Scholar
Voevodsky, V. (2010) The equivalence axiom and univalent models of type theory. Talk at CMU. Available at: http://www.math.ias.edu/~vladimir/Site3/home_files/CMU_talk.pdf, 9 pages.Google Scholar
Warren, M. (2008) Homotopy theoretic Aspects of Constructive Type Theory, Ph.D. thesis, Carnegie Mellon University.Google Scholar